Yacc: Yet Another Compiler-Compiler

ABSTRACT

Computer program input generally has some structure;
in fact, every computer program that does input
can be thought of as defining an ``input language''
which it accepts. An input language may be as complex
as a programming language, or as simple as a sequence
of numbers. Unfortunately, usual input facilities are
limited, difficult to use, and often are lax about
checking their inputs for validity.

Yacc provides a general tool for describing the
input to a computer program. The Yacc user specifies
the structures of his input, together with code to be
invoked as each such structure is recognized. Yacc
turns such a specification into a subroutine that handles
the input process; frequently, it is convenient
and appropriate to have most of the flow of control in
the user's application handled by this subroutine.

The input subroutine produced by Yacc calls a
user-supplied routine to return the next basic input
item. Thus, the user can specify his input in terms of
individual input characters, or in terms of higher
level constructs such as names and numbers. The user-supplied
routine may also handle idiomatic features
such as comment and continuation conventions, which
typically defy easy grammatical specification.

Yacc is written in portable C. The class of
specifications accepted is a very general one: LALR(1)
grammars with disambiguating rules.

In addition to compilers for C, APL, Pascal, RATFOR,
etc., Yacc has also been used for less conventional
languages, including a phototypesetter language,
several desk calculator languages, a document retrieval
system, and a Fortran debugging system.

0: Introduction

Yacc provides a general tool for imposing structure on the
input to a computer program. The Yacc user prepares a specification
of the input process; this includes rules describing the
input structure, code to be invoked when these rules are recognized,
and a low-level routine to do the basic input. Yacc then
generates a function to control the input process. This function,
called a parser, calls the user-supplied low-level input
routine (the lexical analyzer) to pick up the basic items (called
tokens) from the input stream. These tokens are organized
according to the input structure rules, called grammar rules;
when one of these rules has been recognized, then user code supplied
for this rule, an action, is invoked; actions have the
ability to return values and make use of the values of other
actions.

Yacc is written in a portable dialect of C[1] and the
actions, and output subroutine, are in C as well. Moreover, many
of the syntactic conventions of Yacc follow C.

The heart of the input specification is a collection of
grammar rules. Each rule describes an allowable structure and
gives it a name. For example, one grammar rule might be

date : month_name day ',' year ;

Here, date, month_name, day, and year represent structures of
interest in the input process; presumably, month_name, day, and
year are defined elsewhere. The comma ``,'' is enclosed in single
quotes; this implies that the comma is to appear literally in
the input. The colon and semicolon merely serve as punctuation
in the rule, and have no significance in controlling the input.
Thus, with proper definitions, the input

July 4, 1776

might be matched by the above rule.

An important part of the input process is carried out by the
lexical analyzer. This user routine reads the input stream,
recognizing the lower level structures, and communicates these
tokens to the parser. For historical reasons, a structure recognized
by the lexical analyzer is called a terminal symbol, while
the structure recognized by the parser is called a nonterminal
symbol. To avoid confusion, terminal symbols will usually be
referred to as tokens.

There is considerable leeway in deciding whether to recognize
structures using the lexical analyzer or grammar rules. For
example, the rules

might be used in the above example. The lexical analyzer would
only need to recognize individual letters, and month_name would
be a nonterminal symbol. Such low-level rules tend to waste time
and space, and may complicate the specification beyond Yacc's
ability to deal with it. Usually, the lexical analyzer would
recognize the month names, and return an indication that a
month_name was seen; in this case, month_name would be a token.

Literal characters such as ``,'' must also be passed through
the lexical analyzer, and are also considered tokens.

Specification files are very flexible. It is realively easy
to add to the above example the rule

date : month '/' day '/' year ;

allowing

7 / 4 / 1776

as a synonym for

July 4, 1776

In most cases, this new rule could be ``slipped in'' to a working
system with minimal effort, and little danger of disrupting
existing input.

The input being read may not conform to the specifications.
These input errors are detected as early as is theoretically possible
with a left-to-right scan; thus, not only is the chance of
reading and computing with bad input data substantially reduced,
but the bad data can usually be quickly found. Error handling,
provided as part of the input specifications, permits the reentry
of bad data, or the continuation of the input process after skipping
over the bad data.

In some cases, Yacc fails to produce a parser when given a
set of specifications. For example, the specifications may be
self contradictory, or they may require a more powerful recognition
mechanism than that available to Yacc. The former cases
represent design errors; the latter cases can often be corrected
by making the lexical analyzer more powerful, or by rewriting
some of the grammar rules. While Yacc cannot handle all possible
specifications, its power compares favorably with similar systems;
moreover, the constructions which are difficult for Yacc to

handle are also frequently difficult for human beings to handle.
Some users have reported that the discipline of formulating valid
Yacc specifications for their input revealed errors of conception
or design early in the program development.

The theory underlying Yacc has been described elsewhere.[2,
3, 4] Yacc has been extensively used in numerous practical applications,
including lint,[5] the Portable C Compiler,[6] and a
system for typesetting mathematics.[7]

The next several sections describe the basic process of
preparing a Yacc specification; Section 1 describes the preparation
of grammar rules, Section 2 the preparation of the user supplied
actions associated with these rules, and Section 3 the
preparation of lexical analyzers. Section 4 describes the operation
of the parser. Section 5 discusses various reasons why Yacc
may be unable to produce a parser from a specification, and what
to do about it. Section 6 describes a simple mechanism for handling
operator precedences in arithmetic expressions. Section 7
discusses error detection and recovery. Section 8 discusses the
operating environment and special features of the parsers Yacc
produces. Section 9 gives some suggestions which should improve
the style and efficiency of the specifications. Section 10
discusses some advanced topics, and Section 11 gives acknowledgements.
Appendix A has a brief example, and Appendix B gives a
summary of the Yacc input syntax. Appendix C gives an example
using some of the more advanced features of Yacc, and, finally,
Appendix D describes mechanisms and syntax no longer actively
supported, but provided for historical continuity with older versions
of Yacc.

1: Basic Specifications

Names refer to either tokens or nonterminal symbols. Yacc
requires token names to be declared as such. In addition, for
reasons discussed in Section 3, it is often desirable to include
the lexical analyzer as part of the specification file; it may be
useful to include other programs as well. Thus, every specification
file consists of three sections: the declarations, (grammar)
rules, and programs. The sections are separated by double percent
``%%'' marks. (The percent ``%'' is generally used in Yacc
specifications as an escape character.)

In other words, a full specification file looks like

declarations
%%
rules
%%
programs

The declaration section may be empty. Moreover, if the programs
section is omitted, the second %% mark may be omitted also;

thus, the smallest legal Yacc specification is

%%
rules

Blanks, tabs, and newlines are ignored except that they may
not appear in names or multi-character reserved symbols. Comments
may appear wherever a name is legal; they are enclosed in
/* . . . */, as in C and PL/I.

The rules section is made up of one or more grammar rules.
A grammar rule has the form:

A : BODY ;

A represents a nonterminal name, and BODY represents a sequence
of zero or more names and literals. The colon and the semicolon
are Yacc punctuation.

Names may be of arbitrary length, and may be made up of
letters, dot ``.'', underscore ``_'', and non-initial digits.
Upper and lower case letters are distinct. The names used in the
body of a grammar rule may represent tokens or nonterminal symbols.

A literal consists of a character enclosed in single quotes
``'''. As in C, the backslash ``\'' is an escape character
within literals, and all the C escapes are recognized. Thus

For a number of technical reasons, the NUL character ('\0' or 0)
should never be used in grammar rules.

If there are several grammar rules with the same left hand
side, the vertical bar ``|'' can be used to avoid rewriting the
left hand side. In addition, the semicolon at the end of a rule
can be dropped before a vertical bar. Thus the grammar rules

A : B C D ;
A : E F ;
A : G ;

can be given to Yacc as

A : B C D
| E F
| G
;

It is not necessary that all grammar rules with the same left
side appear together in the grammar rules section, although it
makes the input much more readable, and easier to change.

If a nonterminal symbol matches the empty string, this can
be indicated in the obvious way:

empty : ;

Names representing tokens must be declared; this is most
simply done by writing

%token name1 name2 . . .

in the declarations section. (See Sections 3 , 5, and 6 for much
more discussion). Every name not defined in the declarations
section is assumed to represent a nonterminal symbol. Every nonterminal
symbol must appear on the left side of at least one
rule.

Of all the nonterminal symbols, one, called the start symbol,
has particular importance. The parser is designed to recognize
the start symbol; thus, this symbol represents the largest,
most general structure described by the grammar rules. By
default, the start symbol is taken to be the left hand side of
the first grammar rule in the rules section. It is possible, and
in fact desirable, to declare the start symbol explicitly in the
declarations section using the %start keyword:

%start symbol

The end of the input to the parser is signaled by a special
token, called the endmarker. If the tokens up to, but not
including, the endmarker form a structure which matches the start
symbol, the parser function returns to its caller after the endmarker
is seen; it accepts the input. If the endmarker is seen
in any other context, it is an error.

It is the job of the user-supplied lexical analyzer to
return the endmarker when appropriate; see section 3, below.
Usually the endmarker represents some reasonably obvious I/O
status, such as ``end-of-file'' or ``end-of-record''.

2: Actions

With each grammar rule, the user may associate actions to be

performed each time the rule is recognized in the input process.
These actions may return values, and may obtain the values
returned by previous actions. Moreover, the lexical analyzer can
return values for tokens, if desired.

An action is an arbitrary C statement, and as such can do
input and output, call subprograms, and alter external vectors
and variables. An action is specified by one or more statements,
enclosed in curly braces ``{'' and ``}''. For example,

A : '(' B ')'
{ hello( 1, "abc" ); }

and

XXX : YYY ZZZ
{ printf("a message\n");
flag = 25; }

are grammar rules with actions.

To facilitate easy communication between the actions and the
parser, the action statements are altered slightly. The symbol
``dollar sign'' ``$'' is used as a signal to Yacc in this context.

To return a value, the action normally sets the pseudovariable
``$$'' to some value. For example, an action that does
nothing but return the value 1 is

{ $$ = 1; }

To obtain the values returned by previous actions and the
lexical analyzer, the action may use the pseudo-variables $1, $2,
. . ., which refer to the values returned by the components of
the right side of a rule, reading from left to right. Thus, if
the rule is

A : B C D ;

for example, then $2 has the value returned by C, and $3 the
value returned by D.

As a more concrete example, consider the rule

expr : '(' expr ')' ;

The value returned by this rule is usually the value of the expr
in parentheses. This can be indicated by

expr : '(' expr ')' { $$ = $2 ; }

By default, the value of a rule is the value of the first
element in it ($1). Thus, grammar rules of the form

A : B ;

frequently need not have an explicit action.

In the examples above, all the actions came at the end of
their rules. Sometimes, it is desirable to get control before a
rule is fully parsed. Yacc permits an action to be written in
the middle of a rule as well as at the end. This rule is assumed
to return a value, accessible through the usual mechanism by the
actions to the right of it. In turn, it may access the values
returned by the symbols to its left. Thus, in the rule

A : B
{ $$ = 1; }
C
{ x = $2; y = $3; }
;

the effect is to set x to 1, and y to the value returned by C.

Actions that do not terminate a rule are actually handled by
Yacc by manufacturing a new nonterminal symbol name, and a new
rule matching this name to the empty string. The interior action
is the action triggered off by recognizing this added rule. Yacc
actually treats the above example as if it had been written:

$ACT : /* empty */
{ $$ = 1; }
;
A : B $ACT C
{ x = $2; y = $3; }
;

In many applications, output is not done directly by the
actions; rather, a data structure, such as a parse tree, is constructed
in memory, and transformations are applied to it before
output is generated. Parse trees are particularly easy to construct,
given routines to build and maintain the tree structure
desired. For example, suppose there is a C function node, written
so that the call

node( L, n1, n2 )

creates a node with label L, and descendants n1 and n2, and
returns the index of the newly created node. Then parse tree can
be built by supplying actions such as:

expr : expr '+' expr
{ $$ = node( '+', $1, $3 ); }

in the specification.

The user may define other variables to be used by the
actions. Declarations and definitions can appear in the declarations
section, enclosed in the marks ``%{'' and ``%}''. These
declarations and definitions have global scope, so they are known
to the action statements and the lexical analyzer. For example,

%{ int variable = 0; %}

could be placed in the declarations section, making variable
accessible to all of the actions. The Yacc parser uses only
names beginning in ``yy''; the user should avoid such names.

In these examples, all the values are integers: a discussion
of values of other types will be found in Section 10.

3: Lexical Analysis

The user must supply a lexical analyzer to read the input
stream and communicate tokens (with values, if desired) to the
parser. The lexical analyzer is an integer-valued function
called yylex. The function returns an integer, the token number,
representing the kind of token read. If there is a value associated
with that token, it should be assigned to the external variable
yylval.

The parser and the lexical analyzer must agree on these
token numbers in order for communication between them to take
place. The numbers may be chosen by Yacc, or chosen by the user.
In either case, the ``# define'' mechanism of C is used to allow
the lexical analyzer to return these numbers symbolically. For
example, suppose that the token name DIGIT has been defined in
the declarations section of the Yacc specification file. The
relevant portion of the lexical analyzer might look like:

The intent is to return a token number of DIGIT, and a value
equal to the numerical value of the digit. Provided that the
lexical analyzer code is placed in the programs section of the
specification file, the identifier DIGIT will be defined as the
token number associated with the token DIGIT.

This mechanism leads to clear, easily modified lexical
analyzers; the only pitfall is the need to avoid using any token
names in the grammar that are reserved or significant in C or the
parser; for example, the use of token names if or while will
almost certainly cause severe difficulties when the lexical
analyzer is compiled. The token name error is reserved for error
handling, and should not be used naively (see Section 7).

As mentioned above, the token numbers may be chosen by Yacc
or by the user. In the default situation, the numbers are chosen
by Yacc. The default token number for a literal character is the
numerical value of the character in the local character set.
Other names are assigned token numbers starting at 257.

To assign a token number to a token (including literals),
the first appearance of the token name or literal in the declarations
section can be immediately followed by a nonnegative
integer. This integer is taken to be the token number of the
name or literal. Names and literals not defined by this mechanism
retain their default definition. It is important that all
token numbers be distinct.

For historical reasons, the endmarker must have token number
0 or negative. This token number cannot be redefined by the
user; thus, all lexical analyzers should be prepared to return 0
or negative as a token number upon reaching the end of their
input.

A very useful tool for constructing lexical analyzers is the
Lex program developed by Mike Lesk.[8] These lexical analyzers
are designed to work in close harmony with Yacc parsers. The
specifications for these lexical analyzers use regular expressions
instead of grammar rules. Lex can be easily used to produce
quite complicated lexical analyzers, but there remain some
languages (such as FORTRAN) which do not fit any theoretical
framework, and whose lexical analyzers must be crafted by hand.

4: How the Parser Works

Yacc turns the specification file into a C program, which
parses the input according to the specification given. The algorithm
used to go from the specification to the parser is complex,
and will not be discussed here (see the references for more
information). The parser itself, however, is relatively simple,
and understanding how it works, while not strictly necessary,
will nevertheless make treatment of error recovery and ambiguities
much more comprehensible.

The parser produced by Yacc consists of a finite state
machine with a stack. The parser is also capable of reading and
remembering the next input token (called the lookahead token).
The current state is always the one on the top of the stack. The
states of the finite state machine are given small integer
labels; initially, the machine is in state 0, the stack contains
only state 0, and no lookahead token has been read.

The machine has only four actions available to it, called
shift, reduce, accept, and error. A move of the parser is done
as follows:

1. Based on its current state, the parser decides whether it
needs a lookahead token to decide what action should be
done; if it needs one, and does not have one, it calls yylex
to obtain the next token.

2. Using the current state, and the lookahead token if needed,
the parser decides on its next action, and carries it out.
This may result in states being pushed onto the stack, or
popped off of the stack, and in the lookahead token being
processed or left alone.

The shift action is the most common action the parser takes.
Whenever a shift action is taken, there is always a lookahead
token. For example, in state 56 there may be an action:

IF shift 34

which says, in state 56, if the lookahead token is IF, the
current state (56) is pushed down on the stack, and state 34
becomes the current state (on the top of the stack). The lookahead
token is cleared.

The reduce action keeps the stack from growing without
bounds. Reduce actions are appropriate when the parser has seen
the right hand side of a grammar rule, and is prepared to
announce that it has seen an instance of the rule, replacing the
right hand side by the left hand side. It may be necessary to
consult the lookahead token to decide whether to reduce, but usually
it is not; in fact, the default action (represented by a
``.'') is often a reduce action.

Reduce actions are associated with individual grammar rules.
Grammar rules are also given small integer numbers, leading to
some confusion. The action

. reduce 18

refers to grammar rule 18, while the action

IF shift 34

refers to state 34.

Suppose the rule being reduced is
A : x y z ;

The reduce action depends on the left hand symbol (A in this
case), and the number of symbols on the right hand side (three in
this case). To reduce, first pop off the top three states from
the stack (In general, the number of states popped equals the
number of symbols on the right side of the rule). In effect,
these states were the ones put on the stack while recognizing x,
y, and z, and no longer serve any useful purpose. After popping
these states, a state is uncovered which was the state the parser
was in before beginning to process the rule. Using this
uncovered state, and the symbol on the left side of the rule,
perform what is in effect a shift of A. A new state is obtained,
pushed onto the stack, and parsing continues. There are significant
differences between the processing of the left hand symbol
and an ordinary shift of a token, however, so this action is
called a goto action. In particular, the lookahead token is
cleared by a shift, and is not affected by a goto. In any case,
the uncovered state contains an entry such as:

A goto 20

causing state 20 to be pushed onto the stack, and become the
current state.

In effect, the reduce action ``turns back the clock'' in the
parse, popping the states off the stack to go back to the state
where the right hand side of the rule was first seen. The parser
then behaves as if it had seen the left side at that time. If
the right hand side of the rule is empty, no states are popped
off of the stack: the uncovered state is in fact the current
state.

The reduce action is also important in the treatment of
user-supplied actions and values. When a rule is reduced, the
code supplied with the rule is executed before the stack is
adjusted. In addition to the stack holding the states, another
stack, running in parallel with it, holds the values returned
from the lexical analyzer and the actions. When a shift takes
place, the external variable yylval is copied onto the value
stack. After the return from the user code, the reduction is
carried out. When the goto action is done, the external variable
yyval is copied onto the value stack. The pseudo-variables $1,
$2, etc., refer to the value stack.

The other two parser actions are conceptually much simpler.
The accept action indicates that the entire input has been seen
and that it matches the specification. This action appears only
when the lookahead token is the endmarker, and indicates that the
parser has successfully done its job. The error action, on the
other hand, represents a place where the parser can no longer
continue parsing according to the specification. The input

tokens it has seen, together with the lookahead token, cannot be
followed by anything that would result in a legal input. The
parser reports an error, and attempts to recover the situation
and resume parsing: the error recovery (as opposed to the detection
of error) will be covered in Section 7.

When Yacc is invoked with the -v option, a file called
y.output is produced, with a human-readable description of the
parser. The y.output file corresponding to the above grammar
(with some statistics stripped off the end) is:

Notice that, in addition to the actions for each state, there is
a description of the parsing rules being processed in each state.
The _ character is used to indicate what has been seen, and what
is yet to come, in each rule. Suppose the input is

DING DONG DELL

It is instructive to follow the steps of the parser while processing
this input.

Initially, the current state is state 0. The parser needs
to refer to the input in order to decide between the actions
available in state 0, so the first token, DING, is read, becoming
the lookahead token. The action in state 0 on DING is is ``shift
3'', so state 3 is pushed onto the stack, and the lookahead token
is cleared. State 3 becomes the current state. The next token,
DONG, is read, becoming the lookahead token. The action in state
3 on the token DONG is ``shift 6'', so state 6 is pushed onto the
stack, and the lookahead is cleared. The stack now contains 0,
3, and 6. In state 6, without even consulting the lookahead, the
parser reduces by rule 2.

sound : DING DONG

This rule has two symbols on the right hand side, so two states,
6 and 3, are popped off of the stack, uncovering state 0. Consulting
the description of state 0, looking for a goto on sound,

sound goto 2

is obtained; thus state 2 is pushed onto the stack, becoming the
current state.

In state 2, the next token, DELL, must be read. The action
is ``shift 5'', so state 5 is pushed onto the stack, which now
has 0, 2, and 5 on it, and the lookahead token is cleared. In
state 5, the only action is to reduce by rule 3. This has one
symbol on the right hand side, so one state, 5, is popped off,
and state 2 is uncovered. The goto in state 2 on place, the left
side of rule 3, is state 4. Now, the stack contains 0, 2, and 4.
In state 4, the only action is to reduce by rule 1. There are
two symbols on the right, so the top two states are popped off,
uncovering state 0 again. In state 0, there is a goto on rhyme
causing the parser to enter state 1. In state 1, the input is
read; the endmarker is obtained, indicated by ``$end'' in the
y.output file. The action in state 1 when the endmarker is seen
is to accept, successfully ending the parse.

The reader is urged to consider how the parser works when
confronted with such incorrect strings as DING DONG DONG, DING
DONG, DING DONG DELL DELL, etc. A few minutes spend with this
and other simple examples will probably be repaid when problems
arise in more complicated contexts.

5: Ambiguity and Conflicts

A set of grammar rules is ambiguous if there is some input
string that can be structured in two or more different ways. For
example, the grammar rule

expr : expr '-' expr

is a natural way of expressing the fact that one way of forming
an arithmetic expression is to put two other expressions together

with a minus sign between them. Unfortunately, this grammar rule
does not completely specify the way that all complex inputs
should be structured. For example, if the input is

expr - expr - expr

the rule allows this input to be structured as either

( expr - expr ) - expr

or as

expr - ( expr - expr )

(The first is called left association, the second right association).

Yacc detects such ambiguities when it is attempting to build
the parser. It is instructive to consider the problem that confronts
the parser when it is given an input such as

expr - expr - expr

When the parser has read the second expr, the input that it has
seen:

expr - expr

matches the right side of the grammar rule above. The parser
could reduce the input by applying this rule; after applying the
rule; the input is reduced to expr(the left side of the rule).
The parser would then read the final part of the input:

- expr

and again reduce. The effect of this is to take the left associative
interpretation.

Alternatively, when the parser has seen

expr - expr

it could defer the immediate application of the rule, and continue
reading the input until it had seen

expr - expr - expr

It could then apply the rule to the rightmost three symbols,
reducing them to expr and leaving

expr - expr

Now the rule can be reduced once more; the effect is to take the
right associative interpretation. Thus, having read

expr - expr

the parser can do two legal things, a shift or a reduction, and
has no way of deciding between them. This is called a shift /
reduce conflict. It may also happen that the parser has a choice
of two legal reductions; this is called a reduce / reduce conflict.
Note that there are never any ``Shift/shift'' conflicts.

When there are shift/reduce or reduce/reduce conflicts, Yacc
still produces a parser. It does this by selecting one of the
valid steps wherever it has a choice. A rule describing which
choice to make in a given situation is called a disambiguating
rule.

Yacc invokes two disambiguating rules by default:

1. In a shift/reduce conflict, the default is to do the shift.

2. In a reduce/reduce conflict, the default is to reduce by the
earlier grammar rule (in the input sequence).

Rule 1 implies that reductions are deferred whenever there
is a choice, in favor of shifts. Rule 2 gives the user rather
crude control over the behavior of the parser in this situation,
but reduce/reduce conflicts should be avoided whenever possible.

Conflicts may arise because of mistakes in input or logic,
or because the grammar rules, while consistent, require a more
complex parser than Yacc can construct. The use of actions
within rules can also cause conflicts, if the action must be done
before the parser can be sure which rule is being recognized. In
these cases, the application of disambiguating rules is inappropriate,
and leads to an incorrect parser. For this reason,
Yacc always reports the number of shift/reduce and reduce/reduce
conflicts resolved by Rule 1 and Rule 2.

In general, whenever it is possible to apply disambiguating
rules to produce a correct parser, it is also possible to rewrite
the grammar rules so that the same inputs are read but there are
no conflicts. For this reason, most previous parser generators
have considered conflicts to be fatal errors. Our experience has
suggested that this rewriting is somewhat unnatural, and produces
slower parsers; thus, Yacc will produce parsers even in the presence
of conflicts.

As an example of the power of disambiguating rules, consider
a fragment from a programming language involving an ``if-then-else''
construction:

stat : IF '(' cond ')' stat
| IF '(' cond ')' stat ELSE stat
;

In these rules, IF and ELSE are tokens, cond is a nonterminal
symbol describing conditional (logical) expressions, and stat is
a nonterminal symbol describing statements. The first rule will
be called the simple-if rule, and the second the if-else rule.

These two rules form an ambiguous construction, since input
of the form

IF ( C1 ) IF ( C2 ) S1 ELSE S2

can be structured according to these rules in two ways:

IF ( C1 ) {
IF ( C2 ) S1
}
ELSE S2

or

IF ( C1 ) {
IF ( C2 ) S1
ELSE S2
}

The second interpretation is the one given in most programming
languages having this construct. Each ELSE is associated with
the last preceding ``un-ELSE'd'' IF. In this example, consider
the situation where the parser has seen

IF ( C1 ) IF ( C2 ) S1

and is looking at the ELSE. It can immediately reduce by the
simple-if rule to get

IF ( C1 ) stat

and then read the remaining input,

ELSE S2

and reduce

IF ( C1 ) stat ELSE S2

by the if-else rule. This leads to the first of the above groupings
of the input.

On the other hand, the ELSE may be shifted, S2 read, and
then the right hand portion of

IF ( C1 ) IF ( C2 ) S1 ELSE S2

can be reduced by the if-else rule to get

IF ( C1 ) stat

which can be reduced by the simple-if rule. This leads to the
second of the above groupings of the input, which is usually
desired.

Once again the parser can do two valid things - there is a
shift/reduce conflict. The application of disambiguating rule 1
tells the parser to shift in this case, which leads to the
desired grouping.

This shift/reduce conflict arises only when there is a particular
current input symbol, ELSE, and particular inputs already
seen, such as

IF ( C1 ) IF ( C2 ) S1

In general, there may be many conflicts, and each one will be
associated with an input symbol and a set of previously read
inputs. The previously read inputs are characterized by the
state of the parser.

The conflict messages of Yacc are best understood by examining
the verbose (-v) option output file. For example, the output
corresponding to the above conflict state might be:

23: shift/reduce conflict (shift 45, reduce 18) on ELSE

state 23

stat : IF ( cond ) stat_ (18)
stat : IF ( cond ) stat_ELSE stat

ELSE shift 45
. reduce 18

The first line describes the conflict, giving the state and the
input symbol. The ordinary state description follows, giving the
grammar rules active in the state, and the parser actions.
Recall that the underline marks the portion of the grammar rules
which has been seen. Thus in the example, in state 23 the parser
has seen input corresponding to

IF ( cond ) stat

and the two grammar rules shown are active at this time. The
parser can do two possible things. If the input symbol is ELSE,
it is possible to shift into state 45. State 45 will have, as
part of its description, the line

stat : IF ( cond ) stat ELSE_stat

since the ELSE will have been shifted in this state. Back in
state 23, the alternative action, described by ``.'', is to be
done if the input symbol is not mentioned explicitly in the above
actions; thus, in this case, if the input symbol is not ELSE, the
parser reduces by grammar rule 18:

stat : IF '(' cond ')' stat

Once again, notice that the numbers following ``shift'' commands
refer to other states, while the numbers following ``reduce''
commands refer to grammar rule numbers. In the y.output file,
the rule numbers are printed after those rules which can be
reduced. In most one states, there will be at most reduce action
possible in the state, and this will be the default command. The
user who encounters unexpected shift/reduce conflicts will probably
want to look at the verbose output to decide whether the
default actions are appropriate. In really tough cases, the user
might need to know more about the behavior and construction of
the parser than can be covered here. In this case, one of the
theoretical references[2, 3, 4] might be consulted; the services
of a local guru might also be appropriate.

6: Precedence

There is one common situation where the rules given above
for resolving conflicts are not sufficient; this is in the parsing
of arithmetic expressions. Most of the commonly used constructions
for arithmetic expressions can be naturally described
by the notion of precedence levels for operators, together with
information about left or right associativity. It turns out that
ambiguous grammars with appropriate disambiguating rules can be
used to create parsers that are faster and easier to write than
parsers constructed from unambiguous grammars. The basic notion
is to write grammar rules of the form

expr : expr OP expr

and

expr : UNARY expr

for all binary and unary operators desired. This creates a very
ambiguous grammar, with many parsing conflicts. As disambiguating
rules, the user specifies the precedence, or binding
strength, of all the operators, and the associativity of the
binary operators. This information is sufficient to allow Yacc
to resolve the parsing conflicts in accordance with these rules,
and construct a parser that realizes the desired precedences and
associativities.

The precedences and associativities are attached to tokens
in the declarations section. This is done by a series of lines
beginning with a Yacc keyword: %left, %right, or %nonassoc, followed
by a list of tokens. All of the tokens on the same line

are assumed to have the same precedence level and associativity;
the lines are listed in order of increasing precedence or binding
strength. Thus,

%left '+' '-'
%left '*' '/'

describes the precedence and associativity of the four arithmetic
operators. Plus and minus are left associative, and have lower
precedence than star and slash, which are also left associative.
The keyword %right is used to describe right associative operators,
and the keyword %nonassoc is used to describe operators,
like the operator .LT. in Fortran, that may not associate with
themselves; thus,

A .LT. B .LT. C

is illegal in Fortran, and such an operator would be described
with the keyword %nonassoc in Yacc. As an example of the
behavior of these declarations, the description

When this mechanism is used, unary operators must, in general, be
given a precedence. Sometimes a unary operator and a binary
operator have the same symbolic representation, but different
precedences. An example is unary and binary '-'; unary minus may
be given the same strength as multiplication, or even higher,
while binary minus has a lower strength than multiplication. The
keyword, %prec, changes the precedence level associated with a
particular grammar rule. %prec appears immediately after the
body of the grammar rule, before the action or closing semicolon,
and is followed by a token name or literal. It causes the precedence
of the grammar rule to become that of the following token

name or literal. For example, to make unary minus have the same
precedence as multiplication the rules might resemble:

A token declared by %left, %right, and %nonassoc need not
be, but may be, declared by %token as well.

The precedences and associativities are used by Yacc to
resolve parsing conflicts; they give rise to disambiguating
rules. Formally, the rules work as follows:

1. The precedences and associativities are recorded for those
tokens and literals that have them.

2. A precedence and associativity is associated with each grammar
rule; it is the precedence and associativity of the last
token or literal in the body of the rule. If the %prec construction
is used, it overrides this default. Some grammar
rules may have no precedence and associativity associated
with them.

3. When there is a reduce/reduce conflict, or there is a
shift/reduce conflict and either the input symbol or the
grammar rule has no precedence and associativity, then the
two disambiguating rules given at the beginning of the section
are used, and the conflicts are reported.

4. If there is a shift/reduce conflict, and both the grammar
rule and the input character have precedence and associativity
associated with them, then the conflict is resolved
in favor of the action (shift or reduce) associated with the
higher precedence. If the precedences are the same, then
the associativity is used; left associative implies reduce,
right associative implies shift, and nonassociating implies
error.

Conflicts resolved by precedence are not counted in the
number of shift/reduce and reduce/reduce conflicts reported by
Yacc. This means that mistakes in the specification of precedences
may disguise errors in the input grammar; it is a good
idea to be sparing with precedences, and use them in an

essentially ``cookbook'' fashion, until some experience has been
gained. The y.output file is very useful in deciding whether the
parser is actually doing what was intended.

7: Error Handling

Error handling is an extremely difficult area, and many of
the problems are semantic ones. When an error is found, for
example, it may be necessary to reclaim parse tree storage,
delete or alter symbol table entries, and, typically, set
switches to avoid generating any further output.

It is seldom acceptable to stop all processing when an error
is found; it is more useful to continue scanning the input to
find further syntax errors. This leads to the problem of getting
the parser ``restarted'' after an error. A general class of
algorithms to do this involves discarding a number of tokens from
the input string, and attempting to adjust the parser so that
input can continue.

To allow the user some control over this process, Yacc provides
a simple, but reasonably general, feature. The token name
``error'' is reserved for error handling. This name can be used
in grammar rules; in effect, it suggests places where errors are
expected, and recovery might take place. The parser pops its
stack until it enters a state where the token ``error'' is legal.
It then behaves as if the token ``error'' were the current lookahead
token, and performs the action encountered. The lookahead
token is then reset to the token that caused the error. If no
special error rules have been specified, the processing halts
when an error is detected.

In order to prevent a cascade of error messages, the parser,
after detecting an error, remains in error state until three
tokens have been successfully read and shifted. If an error is
detected when the parser is already in error state, no message is
given, and the input token is quietly deleted.

As an example, a rule of the form

stat : error

would, in effect, mean that on a syntax error the parser would
attempt to skip over the statement in which the error was seen.
More precisely, the parser will scan ahead, looking for three
tokens that might legally follow a statement, and start processing
at the first of these; if the beginnings of statements are
not sufficiently distinctive, it may make a false start in the
middle of a statement, and end up reporting a second error where
there is in fact no error.

Actions may be used with these special error rules. These
actions might attempt to reinitialize tables, reclaim symbol
table space, etc.

Error rules such as the above are very general, but difficult
to control. Somewhat easier are rules such as

stat : error ';'

Here, when there is an error, the parser attempts to skip over
the statement, but will do so by skipping to the next ';'. All
tokens after the error and before the next ';' cannot be shifted,
and are discarded. When the ';' is seen, this rule will be
reduced, and any ``cleanup'' action associated with it performed.

Another form of error rule arises in interactive applications,
where it may be desirable to permit a line to be reentered
after an error. A possible error rule might be

There is one potential difficulty with this approach; the parser
must correctly process three input tokens before it admits that
it has correctly resynchronized after the error. If the reentered
line contains an error in the first two tokens, the parser
deletes the offending tokens, and gives no message; this is
clearly unacceptable. For this reason, there is a mechanism that
can be used to force the parser to believe that an error has been
fully recovered from. The statement

yyerrok ;

in an action resets the parser to its normal mode. The last
example is better written

As mentioned above, the token seen immediately after the
``error'' symbol is the input token at which the error was
discovered. Sometimes, this is inappropriate; for example, an
error recovery action might take upon itself the job of finding
the correct place to resume input. In this case, the previous
lookahead token must be cleared. The statement

yyclearin ;

in an action will have this effect. For example, suppose the
action after error were to call some sophisticated resynchronization
routine, supplied by the user, that attempted to advance the
input to the beginning of the next valid statement. After this
routine was called, the next token returned by yylex would

presumably be the first token in a legal statement; the old,
illegal token must be discarded, and the error state reset. This
could be done by a rule like

stat : error
{ resynch();
yyerrok ;
yyclearin ; }
;

These mechanisms are admittedly crude, but do allow for a
simple, fairly effective recovery of the parser from many errors;
moreover, the user can get control to deal with the error actions
required by other portions of the program.

8: The Yacc Environment

When the user inputs a specification to Yacc, the output is
a file of C programs, called y.tab.c on most systems (due to
local file system conventions, the names may differ from installation
to installation). The function produced by Yacc is called
yyparse; it is an integer valued function. When it is called, it
in turn repeatedly calls yylex, the lexical analyzer supplied by
the user (see Section 3) to obtain input tokens. Eventually,
either an error is detected, in which case (if no error recovery
is possible) yyparse returns the value 1, or the lexical analyzer
returns the endmarker token and the parser accepts. In this
case, yyparse returns the value 0.

The user must provide a certain amount of environment for
this parser in order to obtain a working program. For example,
as with every C program, a program called main must be defined,
that eventually calls yyparse. In addition, a routine called
yyerror prints a message when a syntax error is detected.

These two routines must be supplied in one form or another
by the user. To ease the initial effort of using Yacc, a library
has been provided with default versions of main and yyerror. The
name of this library is system dependent; on many systems the
library is accessed by a -ly argument to the loader. To show the
triviality of these default programs, the source is given below:

The argument to yyerror is a string containing an error message,
usually the string ``syntax error''. The average application
will want to do better than this. Ordinarily, the program should
keep track of the input line number, and print it along with the
message when a syntax error is detected. The external integer
variable yychar contains the lookahead token number at the time
the error was detected; this may be of some interest in giving
better diagnostics. Since the main program is probably supplied
by the user (to read arguments, etc.) the Yacc library is useful
only in small projects, or in the earliest stages of larger ones.

The external integer variable yydebug is normally set to 0.
If it is set to a nonzero value, the parser will output a verbose
description of its actions, including a discussion of which input
symbols have been read, and what the parser actions are. Depending
on the operating environment, it may be possible to set this
variable by using a debugging system.

9: Hints for Preparing Specifications

This section contains miscellaneous hints on preparing efficient,
easy to change, and clear specifications. The individual
subsections are more or less independent.

Input Style

It is difficult to provide rules with substantial actions
and still have a readable specification file. The following
style hints owe much to Brian Kernighan.

a. Use all capital letters for token names, all lower case
letters for nonterminal names. This rule comes under the
heading of ``knowing who to blame when things go wrong.''

b. Put grammar rules and actions on separate lines. This
allows either to be changed without an automatic need to
change the other.

c. Put all rules with the same left hand side together. Put
the left hand side in only once, and let all following rules
begin with a vertical bar.

d. Put a semicolon only after the last rule with a given left
hand side, and put the semicolon on a separate line. This
allows new rules to be easily added.

e. Indent rule bodies by two tab stops, and action bodies by
three tab stops.

The example in Appendix A is written following this style,
as are the examples in the text of this paper (where space permits).
The user must make up his own mind about these stylistic
questions; the central problem, however, is to make the rules
visible through the morass of action code.

Left Recursion

The algorithm used by the Yacc parser encourages so called
``left recursive'' grammar rules: rules of the form

name : name rest_of_rule ;

These rules frequently arise when writing specifications of
sequences and lists:

list : item
| list ',' item
;

and

seq : item
| seq item
;

In each of these cases, the first rule will be reduced for the
first item only, and the second rule will be reduced for the
second and all succeeding items.

With right recursive rules, such as

seq : item
| item seq
;

the parser would be a bit bigger, and the items would be seen,
and reduced, from right to left. More seriously, an internal
stack in the parser would be in danger of overflowing if a very
long sequence were read. Thus, the user should use left recursion
wherever reasonable.

It is worth considering whether a sequence with zero elements
has any meaning, and if so, consider writing the sequence
specification with an empty rule:

seq : /* empty */
| seq item
;

Once again, the first rule would always be reduced exactly once,
before the first item was read, and then the second rule would be
reduced once for each item read. Permitting empty sequences
often leads to increased generality. However, conflicts might
arise if Yacc is asked to decide which empty sequence it has
seen, when it hasn't seen enough to know!

Lexical Tie-ins

Some lexical decisions depend on context. For example, the

lexical analyzer might want to delete blanks normally, but not
within quoted strings. Or names might be entered into a symbol
table in declarations, but not in expressions.

One way of handling this situation is to create a global
flag that is examined by the lexical analyzer, and set by
actions. For example, suppose a program consists of 0 or more
declarations, followed by 0 or more statements. Consider:

The flag dflag is now 0 when reading statements, and 1 when reading
declarations, except for the first token in the first statement.
This token must be seen by the parser before it can tell
that the declaration section has ended and the statements have
begun. In many cases, this single token exception does not
affect the lexical scan.

This kind of ``backdoor'' approach can be elaborated to a
noxious degree. Nevertheless, it represents a way of doing some
things that are difficult, if not impossible, to do otherwise.

Reserved Words

Some programming languages permit the user to use words like
``if'', which are normally reserved, as label or variable names,
provided that such use does not conflict with the legal use of
these names in the programming language. This is extremely hard
to do in the framework of Yacc; it is difficult to pass information
to the lexical analyzer telling it ``this instance of `if'
is a keyword, and that instance is a variable''. The user can
make a stab at it, using the mechanism described in the last subsection,
but it is difficult.

A number of ways of making this easier are under advisement.
Until then, it is better that the keywords be reserved; that is,
be forbidden for use as variable names. There are powerful
stylistic reasons for preferring this, anyway.

10: Advanced Topics

This section discusses a number of advanced features of
Yacc.

Simulating Error and Accept in Actions

The parsing actions of error and accept can be simulated in
an action by use of macros YYACCEPT and YYERROR. YYACCEPT causes
yyparse to return the value 0; YYERROR causes the parser to
behave as if the current input symbol had been a syntax error;
yyerror is called, and error recovery takes place. These mechanisms
can be used to simulate parsers with multiple endmarkers or
context-sensitive syntax checking.

Accessing Values in Enclosing Rules.

An action may refer to values returned by actions to the
left of the current rule. The mechanism is simply the same as
with ordinary actions, a dollar sign followed by a digit, but in
this case the digit may be 0 or negative. Consider

In the action following the word CRONE, a check is made that the
preceding token shifted was not YOUNG. Obviously, this is only
possible when a great deal is known about what might precede the
symbol noun in the input. There is also a distinctly unstructured
flavor about this. Nevertheless, at times this mechanism
will save a great deal of trouble, especially when a few combinations
are to be excluded from an otherwise regular structure.

Support for Arbitrary Value Types

By default, the values returned by actions and the lexical
analyzer are integers. Yacc can also support values of other
types, including structures. In addition, Yacc keeps track of
the types, and inserts appropriate union member names so that the
resulting parser will be strictly type checked. The Yacc value
stack (see Section 4) is declared to be a union of the various
types of values desired. The user declares the union, and associates
union member names to each token and nonterminal symbol
having a value. When the value is referenced through a $$ or $n
construction, Yacc will automatically insert the appropriate
union name, so that no unwanted conversions will take place. In
addition, type checking commands such as Lint[5] will be far more
silent.

There are three mechanisms used to provide for this typing.
First, there is a way of defining the union; this must be done by
the user since other programs, notably the lexical analyzer, must
know about the union member names. Second, there is a way of
associating a union member name with tokens and nonterminals.
Finally, there is a mechanism for describing the type of those
few values where Yacc can not easily determine the type.

To declare the union, the user includes in the declaration
section:

%union {
body of union ...
}

This declares the Yacc value stack, and the external variables
yylval and yyval, to have type equal to this union. If Yacc was
invoked with the -d option, the union declaration is copied onto
the y.tab.h file. Alternatively, the union may be declared in a
header file, and a typedef used to define the variable YYSTYPE to
represent this union. Thus, the header file might also have
said:

typedef union {
body of union ...
} YYSTYPE;

The header file must be included in the declarations section, by
use of %{ and %}.

Once YYSTYPE is defined, the union member names must be
associated with the various terminal and nonterminal names. The
construction

< name >

is used to indicate a union member name. If this follows one of
the keywords %token, %left, %right, and %nonassoc, the union

member name is associated with the tokens listed. Thus, saying

%left <optype> '+' '-'

will cause any reference to values returned by these two tokens
to be tagged with the union member name optype. Another keyword,
%type, is used similarly to associate union member names with
nonterminals. Thus, one might say

%type <nodetype> expr stat

There remain a couple of cases where these mechanisms are
insufficient. If there is an action within a rule, the value
returned by this action has no a priori type. Similarly, reference
to left context values (such as $0 - see the previous subsection)
leaves Yacc with no easy way of knowing the type. In
this case, a type can be imposed on the reference by inserting a
union member name, between < and >, immediately after the first
$. An example of this usage is

This syntax has little to recommend it, but the situation arises
rarely.

A sample specification is given in Appendix C. The facilities
in this subsection are not triggered until they are used: in
particular, the use of %type will turn on these mechanisms. When
they are used, there is a fairly strict level of checking. For
example, use of $n or $$ to refer to something with no defined
type is diagnosed. If these facilities are not triggered, the
Yacc value stack is used to hold int's, as was true historically.

11: Acknowledgements

Yacc owes much to a most stimulating collection of users,
who have goaded me beyond my inclination, and frequently beyond
my ability, in their endless search for ``one more feature''.
Their irritating unwillingness to learn how to do things my way
has usually led to my doing things their way; most of the time,
they have been right. B. W. Kernighan, P. J. Plauger, S. I.
Feldman, C. Imagna, M. E. Lesk, and A. Snyder will recognize some
of their ideas in the current version of Yacc. C. B. Haley contributed
to the error recovery algorithm. D. M. Ritchie, B. W.
Kernighan, and M. O. Harris helped translate this document into
English. Al Aho also deserves special credit for bringing the
mountain to Mohammed, and other favors.

This example gives the complete Yacc specification for a
small desk calculator; the desk calculator has 26 registers,
labeled ``a'' through ``z'', and accepts arithmetic expressions
made up of the operators +, -, *, /, % (mod operator), & (bitwise
and), | (bitwise or), and assignment. If an expression at the
top level is an assignment, the value is not printed; otherwise
it is. As in C, an integer that begins with 0 (zero) is assumed
to be octal; otherwise, it is assumed to be decimal.

As an example of a Yacc specification, the desk calculator
does a reasonable job of showing how precedences and ambiguities
are used, and demonstrating simple error recovery. The major
oversimplifications are that the lexical analysis phase is much
simpler than for most applications, and the output is produced
immediately, line by line. Note the way that decimal and octal
integers are read in by the grammar rules; This job is probably
better done by the lexical analyzer.

This Appendix has a description of the Yacc input syntax, as
a Yacc specification. Context dependencies, etc., are not considered.
Ironically, the Yacc input specification language is
most naturally specified as an LR(2) grammar; the sticky part
comes when an identifier is seen in a rule, immediately following
an action. If this identifier is followed by a colon, it is the
start of the next rule; otherwise it is a continuation of the
current rule, which just happens to have an action embedded in
it. As implemented, the lexical analyzer looks ahead after seeing
an identifier, and decide whether the next token (skipping
blanks, newlines, comments, etc.) is a colon. If so, it returns
the token C_IDENTIFIER. Otherwise, it returns IDENTIFIER.
Literals (quoted strings) are also returned as IDENTIFIERS, but
never as part of C_IDENTIFIERs.

This Appendix gives an example of a grammar using some of
the advanced features discussed in Section 10. The desk calculator
example in Appendix A is modified to provide a desk calculator
that does floating point interval arithmetic. The calculator
understands floating point constants, the arithmetic operations
+, -, *, /, unary -, and = (assignment), and has 26 floating
point variables, ``a'' through ``z''. Moreover, it also understands
intervals, written

( x , y )

where x is less than or equal to y. There are 26 interval valued
variables ``A'' through ``Z'' that may also be used. The usage
is similar to that in Appendix A; assignments return no value,
and print nothing, while expressions print the (floating or
interval) value.

This example explores a number of interesting features of
Yacc and C. Intervals are represented by a structure, consisting
of the left and right endpoint values, stored as double's. This
structure is given a type name, INTERVAL, by using typedef. The
Yacc value stack can also contain floating point scalars, and
integers (used to index into the arrays holding the variable
values). Notice that this entire strategy depends strongly on
being able to assign structures and unions in C. In fact, many
of the actions call functions that return structures as well.

It is also worth noting the use of YYERROR to handle error
conditions: division by an interval containing 0, and an interval
presented in the wrong order. In effect, the error recovery
mechanism of Yacc is used to throw away the rest of the offending
line.

In addition to the mixing of types on the value stack, this
grammar also demonstrates an interesting use of syntax to keep
track of the type (e.g. scalar or interval) of intermediate
expressions. Note that a scalar can be automatically promoted to
an interval if the context demands an interval value. This
causes a large number of conflicts when the grammar is run
through Yacc: 18 Shift/Reduce and 26 Reduce/Reduce. The problem
can be seen by looking at the two input lines:

2.5 + ( 3.5 - 4. )

and

2.5 + ( 3.5 , 4. )

Notice that the 2.5 is to be used in an interval valued expression
in the second example, but this fact is not known until the
``,'' is read; by this time, 2.5 is finished, and the parser cannot
go back and change its mind. More generally, it might be

necessary to look ahead an arbitrary number of tokens to decide
whether to convert a scalar to an interval. This problem is
evaded by having two rules for each binary interval valued operator:
one when the left operand is a scalar, and one when the left
operand is an interval. In the second case, the right operand
must be an interval, so the conversion will be applied automatically.
Despite this evasion, there are still many cases where
the conversion may be applied or not, leading to the above conflicts.
They are resolved by listing the rules that yield
scalars first in the specification file; in this way, the conflicts
will be resolved in the direction of keeping scalar valued
expressions scalar valued until they are forced to become intervals.

This way of handling multiple types is very instructive, but
not very general. If there were many kinds of expression types,
instead of just two, the number of rules needed would increase
dramatically, and the conflicts even more dramatically. Thus,
while this example is instructive, it is better practice in a
more normal programming language environment to keep the type
information as part of the value, and not as part of the grammar.

Finally, a word about the lexical analysis. The only
unusual feature is the treatment of floating point constants.
The C library routine atof is used to do the actual conversion
from a character string to a double precision value. If the lexical
analyzer detects an error, it responds by returning a token
that is illegal in the grammar, provoking a syntax error in the
parser, and thence error recovery.

This Appendix mentions synonyms and features which are supported
for historical continuity, but, for various reasons, are
not encouraged.

1. Literals may also be delimited by double quotes ``"''.

2. Literals may be more than one character long. If all the
characters are alphabetic, numeric, or _, the type number of
the literal is defined, just as if the literal did not have
the quotes around it. Otherwise, it is difficult to find
the value for such literals.

The use of multi-character literals is likely to mislead
those unfamiliar with Yacc, since it suggests that Yacc is
doing a job which must be actually done by the lexical
analyzer.

3. Most places where % is legal, backslash ``\'' may be used.
In particular, \\ is the same as %%, \left the same as
%left, etc.

4. There are a number of other synonyms:

%< is the same as %left
%> is the same as %right
%binary and %2 are the same as %nonassoc
%0 and %term are the same as %token
%= is the same as %prec

5. Actions may also have the form

={ . . . }

and the curly braces can be dropped if the action is a single
C statement.

6. C code between %{ and %} used to be permitted at the head of
the rules section, as well as in the declaration section.